Theory Of Computation! Computer Science Trivia

The correct answer is "only regular grammar". A Finite State Machine (FSM) is a computational model that can recognize languages defined by regular grammars. Regular grammars are a subset of formal grammars that can be described using regular expressions or regular language rules. FSMs are not capable of recognizing languages defined by context-free grammars or any other type of grammar. Therefore, the statement "only regular grammar" accurately describes the recognition capabilities of a FSM.

The pumping lemma is a tool used in formal language theory to prove that a given grammar is not regular. It provides a set of conditions that must hold true for all regular languages, and if these conditions are violated, then the language is not regular. By applying the pumping lemma to a given grammar and finding a contradiction, we can conclude that the grammar is not regular.

The set A has two elements, 0 and 1. To form a string of length n, we have 2 choices for each position in the string, either 0 or 1. Therefore, the total number of possible strings is 2 raised to the power of n, which is represented as 2^n.

TM (Turing Machine) is more powerful than FSM (Finite State Machine) because it has the capability to remember arbitrary long sequences of input symbols. Unlike FSM, which has a finite number of states and can only remember a limited amount of information, TM has an unbounded tape that allows it to store and retrieve an infinite amount of data. This ability to remember and manipulate unlimited input sequences gives TM a higher level of computational power, making it more versatile and capable of solving complex problems.

An FSM, or Finite State Machine, is a computational model that can be in one of a finite number of states at any given time. The basic limitation of an FSM is that it can't remember an arbitrary large amount of information. This means that it has a limited memory capacity and can only store a certain amount of information about its past states and inputs. Therefore, it is unable to remember and process a vast amount of data, which is a significant limitation of this model.

The given grammar is classified as context-sensitive because it allows for rules where the length of the right-hand side can be greater than or equal to the length of the left-hand side. Context-sensitive grammars are more powerful than regular grammars and context-free grammars, as they can handle a wider range of languages.

The given regular expressions are all equivalent. In the first pair, both expressions match any string that starts with zero or more occurrences of "ab" followed by an "a". In the second pair, both expressions match any string that consists of zero or more occurrences of "a" or "b". In the third pair, both expressions also match any string that consists of zero or more occurrences of "a" or "b". Therefore, none of the pairs of regular expressions are not equivalent.

The recognizing capability of Nondeterministic Finite State Machines (NDFSM) and Deterministic Finite State Machines (DFSM) must be the same. This means that both types of machines should be able to recognize the same set of languages. While NDFSMs have the ability to have multiple transitions for a single input symbol, DFSMs have a unique transition for each input symbol. However, by converting an NDFSM to a DFSM using methods like the subset construction algorithm, the recognizing capability can be preserved, ensuring that the two types of machines recognize the same languages.

The correct answer states that any string of terminals generated by the given CFG must have at least two 'a's. This means that in order for a string to be valid, it must contain at least two occurrences of the letter 'a'.

The conversion from a non-deterministic pushdown automaton (PDA) to a deterministic PDA is not possible algorithmically. This is because a non-deterministic PDA can have multiple possible transitions for a given input symbol and stack symbol, while a deterministic PDA can only have one. Therefore, it is not possible to determine which transition to take in the deterministic PDA, leading to the impossibility of algorithmically converting a non-deterministic PDA to a deterministic one.

The correct answer is "Complementation" because when we take the complement of a context-free language, the resulting language may not be context-free. Complementation involves swapping the accepting and non-accepting states of a language, which can lead to a change in the structure and properties of the language. Therefore, the complement of a context-free language may not be context-free, making CFG not closed under complementation.

A Universal Turing Machine (UTM) is a theoretical machine that can simulate any other Turing machine. In order to simulate the behavior of any Turing machine, a UTM needs to have at least two internal states. One state is required to represent the current state of the machine being simulated, and the other state is needed to keep track of the current position on the tape. With these two states, a UTM can effectively simulate the operation of any Turing machine, making 2 the minimum number of internal states required.

The FSM in the figure recognizes any string of even number of a's and even number of b's because it starts at state A and transitions to state B when it encounters an 'a', then transitions back to state A when it encounters another 'a'. Similarly, it transitions to state C when it encounters a 'b' and transitions back to state A when it encounters another 'b'. This pattern ensures that the number of 'a's and 'b's in the string is always even.

The intersection of a context-free language (CFL) and a regular language can be context-free. This is because the intersection of these two languages can be recognized by a pushdown automaton, which is a type of automaton that can recognize context-free languages. Therefore, the intersection is always context-free.

The set of regular languages over a given alphabet set is closed under union, complementation, and intersection. This means that when we take the union, complement, or intersection of two regular languages, the resulting language is also regular. Therefore, the correct answer is "none of these" because the set of regular languages is closed under all of these operations.

The given transition graph can be converted into a regular expression, a deterministic finite state machine (DFSM), or a non-deterministic finite state machine (NDFSM). This means that all of these options are correct. Regular expressions represent a pattern of strings that can be accepted by the transition graph. A DFSM is a computational model that can recognize and accept or reject strings based on the transitions in the graph. Similarly, an NDFSM can also recognize and accept or reject strings, but it allows for multiple possible transitions at each state. Therefore, all of these options are valid representations of the given transition graph.

The set A={an bnan | n=1,2,3,...} is not context-free because it cannot be generated by a context-free grammar. In a context-free grammar, the number of a's and b's on either side of the middle symbol (bn) must be equal. However, in this set, the number of a's on the left side is dependent on the value of n, while the number of a's on the right side is dependent on the value of 2n. Therefore, the set A is not context-free.

The given CFG is equivalent to all of the regular expressions mentioned - (a + b)*, (a + b)(a + b)*, and (a + b)*(a + b). This means that any string that can be generated by the CFG can also be generated by any of these regular expressions. Conversely, any string generated by any of these regular expressions can also be generated by the CFG. Therefore, all of these regular expressions are equivalent to the given CFG.

The FSM pictured in the figure is a Mealy Machine because it is a finite state machine that outputs a value based on both its current state and the input. In a Mealy Machine, the outputs are associated with the transitions between states, meaning that the output is determined by the current state and the input that triggers the transition. This is different from a Moore Machine where the outputs are associated with the states themselves. A Kleene Machine is not a recognized term in the context of finite state machines.

The major difference between a Moore machine and a Mealy machine lies in how they handle output. The key distinction is in when the output is produced—whether it's based solely on the current state (Moore) or on both the current state and input (Mealy).

The answer states that any regular language can be generated by a context-free grammar (CFG). This means that for every regular language, there exists a CFG that can generate it. This is a fundamental result in formal language theory, where regular languages are a subset of context-free languages. The statement implies that CFGs are powerful enough to generate regular languages, but it does not imply that all languages can be generated by CFGs.

An FSM (Finite State Machine) can be considered a TM (Turing Machine) of finite tape length, without rewinding capability and unidirectional tape movement. This means that an FSM operates on a tape with a limited length, it cannot rewind the tape, and it can only move the tape in one direction. This is in contrast to a TM with rewinding capability or bidirectional tape movement, which have more flexibility in their tape operations.

The given correct answer is S → ab / aSb. This is because the set {a^nb^n | n=1,2,3......} consists of strings where the number of 'a's is equal to the number of 'b's, and each 'a' is followed by a 'b'. The production rule S → ab / aSb generates strings of this form. The first option, S → aaSbb / ab, generates strings where there are more than one 'a's followed by more than one 'b's, which is not a part of the given set. The third option, S → ab / aSb / ε, includes the possibility of generating empty strings, which is not mentioned in the given set. Therefore, the correct answer is S → ab / aSb.

The given regular expressions "x(xx)*" and "(xx)*x" are equivalent because they both represent strings that start and end with the same character 'x', followed by any number of occurrences of the substring 'xx' in between. The order of the occurrences of 'xx' does not matter, hence the two regular expressions are equivalent.

The language L = {an bnan | n=1,2,3,...} is not context free. This can be proved using the pumping lemma for context-free languages. According to the pumping lemma, if a language is context free, then there exists a constant p (the pumping length) such that any string in the language with length greater than or equal to p can be divided into five parts uvwxy, satisfying certain conditions. However, in the language L, the number of a's and b's are related, and it is not possible to satisfy the conditions of the pumping lemma. Therefore, L is not context free.